Average radius of curvature of set of points on a road?

Question

I have a set of points (lat, lng) on a serpentine road. What would be a good way to calculate average radius of curvature of the road?

Also even without considering (lat, lng) points, just on a 2D surface, assuming there are lots of points (xi, yi) which can be part of a 2D road, what is the best way to calculate 1 - an overall curvature 2 - individual curvatures of each of the convex/concave sections.

One can partition by calculating the inflection points, and then may be use the formula given here but I was wondering any other better way of doing this. Since it is possible to have a convex section which is very complicated and just taking 3 points may not give the full picture.

Answer 1

The proper definition of curvature of a smooth curve involves the second derivative, The wikipedia curvature page has all the formulas. With those you can calculate the curvature at any point, the total curvature and the average curvature (the latter two will be integrals).

If the known points on your curve are close enough together (particularly where the curvature is large) you can calculate the derivatives and integrals numerically.

Answer 2

To achieve this, I would use a smooth interpolating curve with at least $G_2$ continuity. The cubic spline is a good candidate.

Then you can establish the analytical expression of the curvature for every piece of the spline and draw a plot with arbitrary sampling. You will probably want to plot the curvature as a function of the curvilinear abscissa (intrinsic curve equation).

You can obtain the curvilinear abscissas by numerical integration, and at the same time, integrate the (absolute) curvature to get an average.